Properties

Label 2-15e2-1.1-c1-0-3
Degree $2$
Conductor $225$
Sign $-1$
Analytic cond. $1.79663$
Root an. cond. $1.34038$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s − 3·7-s − 2·11-s + 13-s + 6·14-s − 4·16-s − 2·17-s − 5·19-s + 4·22-s − 6·23-s − 2·26-s − 6·28-s − 10·29-s − 3·31-s + 8·32-s + 4·34-s + 2·37-s + 10·38-s + 8·41-s + 43-s − 4·44-s + 12·46-s − 2·47-s + 2·49-s + 2·52-s + 4·53-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 1.13·7-s − 0.603·11-s + 0.277·13-s + 1.60·14-s − 16-s − 0.485·17-s − 1.14·19-s + 0.852·22-s − 1.25·23-s − 0.392·26-s − 1.13·28-s − 1.85·29-s − 0.538·31-s + 1.41·32-s + 0.685·34-s + 0.328·37-s + 1.62·38-s + 1.24·41-s + 0.152·43-s − 0.603·44-s + 1.76·46-s − 0.291·47-s + 2/7·49-s + 0.277·52-s + 0.549·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{225} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + p T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.36506854345797939837793722937, −10.51847852466115917559094220478, −9.725369907410972008022337048795, −8.938558667143995354006242020404, −7.965886472126962800720761127653, −6.96141726621690601574492077524, −5.87930303915993667052597404250, −4.01465426298836833468759021488, −2.23186561512245260821665501913, 0, 2.23186561512245260821665501913, 4.01465426298836833468759021488, 5.87930303915993667052597404250, 6.96141726621690601574492077524, 7.965886472126962800720761127653, 8.938558667143995354006242020404, 9.725369907410972008022337048795, 10.51847852466115917559094220478, 11.36506854345797939837793722937

Graph of the $Z$-function along the critical line